Conjugate beta / interpretation of the "continuous binomial" signal Note: this question has significantly evolved, thanks to inspiring comments by Tim.
Assume there is some "truth" $x\in[0,1]=Beta(1,1)$ that is signaled with some precision. I assume that the resulting posterior distribution (after receiving signal) of $x$ is $Beta(\alpha,\beta)$. To make the signal more explicit, I shall reparametrize the distribution as in this question and this paper (also accounting for uniform prior) with:$\alpha=1+s\phi$, $\beta=1+\phi(1-s)$, where $s\in[0,1]$ is a signal and $\phi$ is explicitly the precision parameter.
Since I know both the prior and the posterior, by Bayes rule, the signal $s$ given $x$ and chosen precision $\phi$ must follow:
$$ 
f(s|x,\phi)=\frac{\Gamma(2+\phi)}{\Gamma(1+s\phi)\Gamma(1+\phi(1-s))}\cdot{x^{s\phi}(1-x)^{\phi(1-s)}}.
$$
For $\phi\in\mathbb{N}$ and $s=k/\phi$ for $k\in\mathbb{N}$ this has a nice binomial interpretation: you make $n$ binomial experiments each with success probability $x$ and your signal is equal to the fraction of successes.
What could be an interpretation of a signal in the general case?  Is there some intuition?  Is this formulation ever used?
 A: Your equations and your nice intuition happens for all exponential families.
I would call your $\phi$ the “observation count” rather than precision since precision is usually the reciprocal of the variance.  (This is because I would interpret your Beta distribution as arising from a Beta-binomial model with $\phi$ Bernoulli observations.)
Your $(\phi s, \phi(1-s))$ are the natural parameters of the Beta distribution.  When you combine the evidence of two distributions while assuming independence, you just take the pointwise product of their densities (and divide out any double-counted information).  This is always the same as adding the natural parameters, which is what you've done.
Since $(\phi s, \phi(1-s))$ are natural parameters, another pair of natural parameters is $(\phi s, \phi)$.  It's not surprising to see that $\phi$ is a natural parameter since in the Beta-binomial model, $\phi$ was the number of observations, and combining evidence should add the number of observations.  In fact, every conjugate prior distribution of an exponential family has this “observation count” as a natural parameter. 
As a contrast, for the normal distribution the natural parameters are $\left(\frac{\mu}{\sigma^2}, -\frac1{2\sigma^2}\right)$.  You can try the same calculations as in your question to verify that taking the pointwise product of densities is tantamount to adding natural parameters.
